AY 16-17 Colloquia
Friday, March 3, 2017; 11:00AM in AG 109: Maria Nogin, Ph.D. and Adnan Sabuwala, Ph.D. (Fresno State)
Magic is fun, exciting, and intriguing. While most magic tricks involve sleight of hand, some are purely mathematical in nature. In this presentation, we will perform a few such tricks (sorry, no rabbits coming out of a hat or birds flying around the room). After showing these tricks, we will do the unthinkable -- reveal our tricks! We will both show how each trick works and explain the mathematics behind it.
No knowledge of advanced mathematics is required to learn these tricks. We invite everyone to come have a magical day of fun and learning.
Thursday, February 16, 2017; 3:00PM in S 145: Moshe Cohen, Ph.D. (Vassar College)
Title: Arrangements of lines: when the combinatorics fails to understand topology
Friday, February 17, 2017; 12:00PM in PB 192: Jesse Wang, Ph.D. (The founder and CEO of Enosh Company)
Wednesday, February 08 2017; 3:00 PM in PB 390: Nathan Williams, Ph.D. (UC Santa Barbara)
Title: Sweeping up Zeta
Abstract: I will discuss three combinatorial problems with the same solution. In 2002, R. Suter defined a striking cyclic symmetry of order n + 1 on the subposet of Young's lattice consisting of the 2n partitions with largest hook at most n. These partitions naturally arise from D. Peterson’s parametrization of abelian ideals of a Borel subalgebra using the affine Weyl group (as told by B. Kostant); the cyclic symmetry comes from the fact that the affine Dynkin diagram in type A is a cycle.
Problem 1 . Describe the orbit structure of Suter’s cyclic symmetry. Now fix a and b relatively prime, and let Dyck(a,b) be the set of lattice paths from (0, 0) to (b, a) that stay above the main diagonal. The sweep map – defined by D. Armstrong, N. Loehr, and G. Warrington – is a map from Dyck(a, b) to itself that rearranges the steps of a path according to the order in which they are encountered by a line of slope a/b sweeping down from above. This map has applications in the theory of Macdonald polynomials and diagonal harmonics – it generalizes the zeta map on Dyck(a,a + 1), which converts between J. Haglund’s statistic bounce and M. Haiman’s dinv.
Problem 2 . Show that the sweep map is a bijection on Dyck(a, b). Finally, consider the following problem of scheduling daily, recurring tasks. The day is divided into m hours, and there are N tasks to be carried out each day (numbered, or prioritized, from 1 to N). For simplicity, suppose that the total amount of time required to carry out all of the tasks is some multiple of m. Each task takes an integer number of hours to complete: a task can take zero hours, but no task takes as much as an entire day. Tasks start on the hour, and cannot be interrupted once started. Tasks can be worked on concurrently, and the starting order for the tasks within the day is specified – the jth task must start before or at the same time as the (j+1)st task. The schedule re- peats every day, so tasks can start at the end of one day and finish at the beginning of the next day.
Problem 3 . With these assumptions, find an assignment of starting hours for the tasks so that the workload throughout the day is constant.
This is based on joint work with Hugh Thomas.
Friday, February 03 2017; 9:00 AM in S139: Morgan Rodgers, Ph.D. (Lake Superior State University)
Title: Special line sets in projective space.
Abstract: In a 1984 paper, Cameron and Liebler introduced classes of lines in projective space PG(3,q) that they termed “special”. These sets of lines, now known as Cameron–Liebler line classes, are defined as having their characteristic vector contained in the row space of A, the point-line incidence matrix of PG(3,q). This purely algebraic characterization has been shown to be equivalent to many interesting geometric properties.
After discussing some of the properties that make Cameron–Liebler line classes so “special”, we will look at some algebraic and computational techniques that have been used to construct examples of these sets. We will also give a generalization of these objects to subspaces in higher dimensional projective spaces, and look as some of the connections to the theory of error-correcting codes.
Click here to see the flyer.
Monday, February 06 2017; 3:00 PM in PB 390: Martha Precup, Ph.D. (Northwestern University)
Title: Springer varieties and Row-Standard Tableaux
Abstract: Springer varieties are geometric objects that play an important role in representation theory. A row-strict tableau is a diagram of boxes filled with the numbers 1,...,n so that each number appears exactly once and the values increase from left to right in each row. In this talk, we’ll define Springer varieties and analyze their structure using row-strict tableaux. We will use these diagrams to compute the Betti numbers of Springer varieties and give a simple formula for doing so in some special cases.
Friday, December 02 2016; 4:00 PM in PB 011 : Micah Chrisman, Ph.D. (Monmouth University)
Title: Calculus, Cobordisms, and Knots
Abstract: In calculus, we learn to sketch curves and surfaces from their critical point data. Besides sketching the graph, you can tell a lot about a surface from its critical points. Is it a sphere, a doughnut? This is related to a contemporary research area in mathematics called knot concordance. Given two knots in three space, R^3, when is it possible to connect them by a smooth tube in the four dimensional space R^3 × I whose ends are the given knots? In this talk, we will look at knot concordance through the combinatorial lens of virtual knots. This point of view, being fairly recent, has many accessible open questions. Throughout the talk, we will introduce a few tricks of the trade so that the listener can begin to think about them! Click here! to see the flyer.
Friday, October 28, 2016 @ 3:00PM in PB 011
Paul Savala, Ph.D. (Whittier College)
What do integrals have to do with prime numbers?
Abstract: Number theory is the branch of mathematics concerned with answering questions about the prime numbers. An example of a question in number theory might be, given some infinite list of numbers, does it contain infinitely many primes?
Analysis is the branch of mathematics concerned with tiny changes. Integrals, derivatives, infinite sums and products are all topics that analysts are concerned with.
At first glance these two branches might seem like polar opposites, but this couldn’t be further from the truth! We‘ll talk about some of the surprising ways in which the techniques you learned in calculus are used to answer questions about the primes. We’ll discuss how mathematicians use techniques from many different fields to answer modern mathematical questions.